It seems that having a non-working reg plate light is now an MoT fail (as of this year). Mine (R12S) hasn't worked for some time and it isn't the bulb.

I haven't yet found a good wiring diagram online to make tracing the fault easier. As far as I can see from a first quick look, the wires from the light disappear into the bowels of the bike, to end up... ...who knows where. Ho hum, out with the test meter.

There's a wire from each indicator and one from the tail light on about 1/2 meter of wiring terminating in a small multi plug - hopefully the problem lies in that short stretch. The plug lies under the rear seat unit of the bike and so could be prone to salt/water ingress. According to the parts fiche pins and plugs are available for repair. Hope that helps